Abstract
This chapter addresses the case of a vector particle produced indirectly from the decay or transformation of a particle having \(J = 0\) (\(\chi_{c0} \to \mathrm{J}/\psi \, \gamma\), \(\mathrm{B} \to \mathrm{J}/\psi \, \mathrm{K}\), \(\mathrm{H} \to \mathrm{Z} \, \gamma\), etc.), \(J = 1\) or \(J = 2\) (\(\chi_{c1,2} \to \mathrm{J}/\psi \, \gamma\), \(\mathrm{Z} \to \mathrm{J}/\psi \, \gamma\), etc.), describing the polarization frame definitions that can be adopted to represent the two-step process, and their respective advantages. We illustrate how several measurement and kinematic conditions can lead to very different observable polarizations of the vector particle, in the special but frequent case where the intermediate decay step remains unobserved and only the final dilepton angular distribution is measured. We also discuss how the peculiar observation, made at the LHC, of almost unpolarized “directly” produced J/\(\psi\) mesons may be understood as the result of a two-step production mechanism.
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References
CMS Collaboration: Measurement of J/\(\psi \) and \(\psi \)(2S) prompt double-differential cross sections in pp collisions at \(\sqrt{s}=7\) TeV. Phys. Rev. Lett. 114, 191802 (2015). https://doi.org/10.1103/PhysRevLett.114.191802, arXiv:1502.04155
CMS Collaboration: Measurements of the \(\varUpsilon \)(1S), \(\varUpsilon \)(2S), and \(\varUpsilon \)(3S) differential cross sections in pp collisions at \(\sqrt{s} = 7\) TeV. Phys. Lett. B 749, 14 (2015). https://doi.org/10.1016/j.physletb.2015.07.037, arXiv:1501.07750
CMS Collaboration: Measurement of quarkonium production cross sections in pp collisions at \(\sqrt{s}=13\) TeV. Phys. Lett. B 780, 251 (2018). https://doi.org/10.1016/j.physletb.2018.02.033, arXiv:1710.11002
ATLAS Collaboration: Measurement of the differential cross-sections of prompt and non-prompt production of \({\rm J} /\psi \) and \(\psi \)(2S) in pp collisions at \(\sqrt{s} = 7\) and 8 TeV with the ATLAS detector. Eur. Phys. J. C 76, 283 (2016). https://doi.org/10.1140/epjc/s10052-016-4050-8, arXiv:1512.03657
ATLAS Collaboration: Measurement of \(\varUpsilon \) production in 7 TeV pp collisions at ATLAS. Phys. Rev. D 87, 052004 (2013). https://doi.org/10.1103/PhysRevD.87.052004, arXiv:1211.7255
LHCb Collaboration: Measurement of \({\rm J} /\psi \) production in pp collisions at \(\sqrt{s}=7\) TeV. Eur. Phys. J. C 71, 1645 (2011). https://doi.org/10.1140/epjc/s10052-011-1645-y, arXiv:1103.0423
LHCb Collaboration: Measurement of \(\varUpsilon \) production in pp collisions at \(\sqrt{s} = 7\) TeV. Eur. Phys. J. C 72, 2025 (2012). https://doi.org/10.1140/epjc/s10052-012-2025-y, arXiv:1202.6579
LHCb Collaboration: Forward production of \(\varUpsilon \) mesons in pp collisions at \(\sqrt{s} = 7\) and 8 TeV. JHEP 11, 103 (2015). https://doi.org/10.1007/JHEP11(2015)103, arXiv:1509.02372
LHCb Collaboration: Production of \({\rm J} /\psi \) and \(\varUpsilon \) mesons in pp collisions at \(\sqrt{s} = 8\) TeV. JHEP 06, 064 (2013). https://doi.org/10.1007/JHEP06(2013)064, arXiv:1304.6977
LHCb Collaboration: Measurement of forward \({\rm J} /\psi \) production cross-sections in pp collisions at \(\sqrt{s} = 13\) TeV. JHEP 10, 172 (2015). https://doi.org/10.1007/JHEP10(2015)172, arXiv:1509.00771. [Erratum: JHEP 05, 063 (2017)]
LHCb Collaboration: Measurement of \(\varUpsilon \) production in pp collisions at \(\sqrt{s} = 13\) TeV. JHEP 07, 134 (2018). https://doi.org/10.1007/JHEP07(2018)134, arXiv:1804.09214. [Erratum: JHEP 05, 076 (2019)]
ATLAS Collaboration: Measurement of \(\chi _{c1}\) and \(\chi _{c2}\) production with \(\sqrt{s} = 7\) TeV pp collisions at ATLAS. JHEP 07, 154 (2014). https://doi.org/10.1007/JHEP07(2014)154, arXiv:1404.7035
CMS Collaboration: Measurement of the relative prompt production rate of \(\chi _{c2}\) and \(\chi _{c1}\) in pp collisions at \(\sqrt{s} = 7\) TeV. Eur. Phys. J. C 72, 2251 (2012). https://doi.org/10.1140/epjc/s10052-012-2251-3, arXiv:1210.0875
CMS Collaboration: Measurement of the production cross section ratio \(\sigma (\chi _{b2}\)(1P))/\(\sigma (\chi _{b1}\)(1P)) in pp collisions at \(\sqrt{s} = 8\) TeV. Phys. Lett. B 743, 383 (2015). https://doi.org/10.1016/j.physletb.2015.02.048, arXiv:1409.5761
LHCb Collaboration: Measurement of the cross-section ratio \(\sigma (\chi _{c2})/\sigma (\chi _{c1})\) for prompt \(\chi _c\) production at \(\sqrt{s} = 7\) TeV. Phys. Lett. B 714, 215 (2012). https://doi.org/10.1016/j.physletb.2012.06.077, arXiv:1202.1080
LHCb Collaboration: Measurement of the relative rate of prompt \(\chi _{c0}\), \(\chi _{c1}\) and \(\chi _{c2}\) production at \(\sqrt{s}=7\) TeV. JHEP 10, 115 (2013). https://doi.org/10.1007/JHEP10(2013)115, arXiv:1307.4285
LHCb Collaboration: Measurement of the ratio of prompt \(\chi _{c}\) to \({\rm J} /\psi \) production in pp collisions at \(\sqrt{s} = 7\) TeV. Phys. Lett. B 718, 431 (2012). https://doi.org/10.1016/j.physletb.2012.10.068, arXiv:1204.1462
LHCb Collaboration: Study of \(\chi _b\) meson production in pp collisions at \(\sqrt{s} = 7\) and 8 TeV and observation of the decay \(\chi _b\)(3P) \(\rightarrow \varUpsilon \)(3S)\(\gamma \). Eur. Phys. J. C 74, 3092 (2014). https://doi.org/10.1140/epjc/s10052-014-3092-z, arXiv:1407.7734
CMS Collaboration: Constraints on the \(\chi _{c1}\) versus \(\chi _{c2}\) polarizations in proton-proton collisions at \(\sqrt{s} = 8\) TeV. Phys. Rev. Lett. 124, 162002 (2020). http://dx.doi.org/10.1103/PhysRevLett.124.162002, arXiv:1912.07706
ATLAS Collaboration: Measurement of the production cross-section of \(\psi \)(2S) \(\rightarrow \) J/\(\psi \) (\(\rightarrow \mu ^{+} \mu ^{-}) \pi ^{+} \pi ^{-}\) in pp collisions at \(\sqrt{s} = 7\) TeV at ATLAS. JHEP 09, 079 (2014). https://doi.org/10.1007/JHEP09(2014)079, arXiv:1407.5532
Faccioli, P., et al.: Quarkonium production at the LHC: a data-driven analysis of remarkably simple experimental patterns. Phys. Lett. B 773, 476 (2017). https://doi.org/10.1016/j.physletb.2017.09.006, arXiv:1702.04208
CDF Collaboration: Measurement of the B\(^+\) total cross section and B\(^+\) differential cross section \({{\rm d }}\sigma /{{\rm d }}p_{\rm T }\) in p\(\bar{\text{p}}\) collisions at \(\sqrt{s} = 1.8\) TeV. Phys. Rev. D 65, 052005 (2002). https://doi.org/10.1103/PhysRevD.65.052005, arXiv:hep-ph/0111359
ATLAS Collaboration: Measurement of the differential cross-section of B\(^{+}\) meson production in pp collisions at \(\sqrt{s} = 7\) TeV at ATLAS. JHEP 10, 042 (2013). https://doi.org/10.1007/JHEP10(2013)042, arXiv:1307.0126
ATLAS Collaboration: Combined measurement of differential and total cross sections in the H \(\rightarrow \gamma \gamma \) and the H \(\rightarrow \) ZZ\(^* \rightarrow 4\ell \) decay channels at \(\sqrt{s} = 13\) TeV with the ATLAS detector. Phys. Lett. B 786, 114 (2018). https://doi.org/10.1016/j.physletb.2018.09.019, arXiv:1805.10197
CMS Collaboration: Measurement of the B\(^+\) production cross section in pp collisions at \(\sqrt{s} = 7\) TeV. Phys. Rev. Lett. 106, 112001 (2011). https://doi.org/10.1103/PhysRevLett.106.112001, arXiv:1101.0131
CMS Collaboration: Measurement and interpretation of differential cross sections for Higgs boson production at \(\sqrt{s} =13\) TeV. Phys. Lett. B 792, 369 (2019). https://doi.org/10.1016/j.physletb.2019.03.059, arXiv:1812.06504
Faccioli, P., Lourenço, C., Araújo, M., Seixas, J.: Universal kinematic scaling as a probe of factorized long-distance effects in high-energy quarkonium production. Eur. Phys. J. C 78, 118 (2018). https://doi.org/10.1140/epjc/s10052-018-5610-x, arXiv:1802.01102
CMS Collaboration: Measurement of the prompt \({\rm J}/\psi \) and \(\psi \)(2S) polarizations in pp collisions at \(\sqrt{s} = 7\) TeV. Phys. Lett. B 727, 381 (2013). https://doi.org/10.1016/j.physletb.2013.10.055, arXiv:1307.6070
LHCb Collaboration: Measurement of \({\rm J /\psi }\) polarization in pp collisions at \(\sqrt{s}=7\) TeV. Eur. Phys. J. C 73, 2631 (2013). https://doi.org/10.1140/epjc/s10052-013-2631-3, arXiv:1307.6379
LHCb Collaboration: Measurement of \(\psi \)(2S) polarisation in pp collisions at \(\sqrt{s}=7\) TeV. Eur. Phys. J. C 74, 2872 (2014). https://doi.org/10.1140/epjc/s10052-014-2872-9, arXiv:1403.1339
ALICE Collaboration: \({\rm J/\psi }\) polarization in pp collisions at \(\sqrt{s}=7\) TeV. Phys. Rev. Lett. 108, 082001 (2012). https://doi.org/10.1103/PhysRevLett.108.082001, arXiv:1111.1630
Faccioli, P., Lourenço, C., Madlener, T.: From prompt to direct \({\rm J /\psi }\) production: new insights on the \(\chi _{c1}\) and \(\chi _{c2}\) polarizations and feed-down contributions from a global-fit analysis of mid-rapidity LHC data. Eur. Phys. J. C 80, 623 (2020). https://doi.org/10.1140/epjc/s10052-020-8201-6, arXiv:2006.15446
Ma, Y.-Q., Wang, K., Chao, K.-T.: \({\rm J /\psi }(\psi ^\prime )\) production at the Tevatron and LHC at \({\cal{O}}(\alpha _s^4v^4)\) in nonrelativistic QCD. Phys. Rev. Lett. 106, 042002 (2011). https://doi.org/10.1103/PhysRevLett.106.042002, arXiv:1009.3655
Butenschön, M., Kniehl, B.A.: Next-to-leading-order tests of NRQCD factorization with \(J/\psi \) yield and polarization. Mod. Phys. Lett. A 28, 1350027 (2013). https://doi.org/10.1142/S0217732313500272, arXiv:1212.2037
Baier, R., Rückl, R.: Hadronic collisions: a quarkonium factory. Z. Phys. C 19, 251 (1983). https://doi.org/10.1007/BF01572254
CDF Collaboration: Quarkonia production at CDF. Nucl. Phys. A 610, 373c (1996). https://doi.org/10.1016/S0375-9474(96)00371-5
CDF Collaboration: \({\rm J /\psi }\) and \(\psi \)(2S) production in p\(\bar{{\rm p}}\) collisions at \(\sqrt{s} = 1.8\) TeV. Phys. Rev. Lett. 79, 572 (1997). https://doi.org/10.1103/PhysRevLett.79.572
Bodwin, G.T., Braaten, E., Lepage, P.: Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium. Phys. Rev. D 51, 1125 (1995). https://doi.org/10.1103/PhysRevD.51.1125, arXiv:hep-ph/9407339. [Erratum: Phys. Rev. D 55, 5853 (1997)]
Lansberg, J.-P.: J/\(\psi \) production at \(\sqrt{s} = 1.96\) and 7 TeV: color-singlet model, NNLO* and polarisation. J. Phys. G 38, 124110 (2011). https://doi.org/10.1088/0954-3899/38/12/124110, arXiv:1107.0292
Ma, Y.-Q., Wang, K., Chao, K.-T.: A complete NLO calculation of the J/\(\psi \) and \(\psi \)(2S) production at hadron colliders. Phys. Rev. D 84, 114001 (2011). https://doi.org/10.1103/PhysRevD.84.114001, arXiv:1012.1030
Butenschön, M., Kniehl, B.A.: Reconciling J/\(\psi \) production at HERA, RHIC, Tevatron, and LHC with NRQCD factorization at next-to-leading order. Phys. Rev. Lett. 106, 022003 (2011). https://doi.org/10.1103/PhysRevLett.106.022003, arXiv:1009.5662
Chao, K.-T., et al.: J/\(\psi \) polarization at hadron colliders in nonrelativistic QCD. Phys. Rev. Lett. 108, 242004 (2012). arXiv:1201.2675
Butenschön, M., Kniehl, B.A.: J/\(\psi \) polarization at Tevatron and LHC: nonrelativistic-QCD factorization at the crossroads. Phys. Rev. Lett. 108, 172002 (2012). https://doi.org/10.1103/PhysRevLett.108.172002, arXiv:1201.1872
Gong, B., Wan, L.-P., Wang, J.-X., Zhang, H.-F.: Polarization for prompt J/\(\psi \) and \(\psi \)(2S) production at the Tevatron and LHC. Phys. Rev. Lett. 110, 042002 (2013). https://doi.org/10.1103/PhysRevLett.110.042002, arXiv:1205.6682
Butenschön, M., Kniehl, B.A.: World data of J/\(\psi \) production consolidate NRQCD factorization at NLO. Phys. Rev. D 84, 051501 (2011). https://doi.org/10.1103/PhysRevD.84.051501, arXiv:1105.0820
Bodwin, G.T., Chung, H.S., Kim, U.-R., Lee, J.: Fragmentation contributions to J/\(\psi \) production at the Tevatron and the LHC. Phys. Rev. Lett. 113, 022001 (2014). https://doi.org/10.1103/PhysRevLett.113.022001, arXiv:1403.3612
Faccioli, P., Lourenço, C., Seixas, J., Wöhri, H.K.: Towards the experimental clarification of quarkonium polarization. Eur. Phys. J. C 69, 657 (2010). https://doi.org/10.1140/epjc/s10052-010-1420-5, arXiv:1006.2738
Faccioli, P., et al.: Quarkonium production in the LHC era: a polarized perspective. Phys. Lett. B 736, 98 (2014). https://doi.org/10.1016/j.physletb.2014.07.006, arXiv:1403.3970
Bodwin, G.T., et al.: Fragmentation contributions to hadroproduction of prompt J/\(\psi \), \(\chi _{cJ}\), and \(\psi \)(2S) states. Phys. Rev. D 93, 034041 (2016). https://doi.org/10.1103/PhysRevD.93.034041, arXiv:1509.07904
Faccioli, P., et al.: From identical S- and P-wave \(p_{\text{ T }}\) spectra to maximally distinct polarizations: probing NRQCD with \(\chi \) states. Eur. Phys. J. C 78, 268 (2018). https://doi.org/10.1140/epjc/s10052-018-5755-7, arXiv:1802.01106
BES Collaboration: \(\psi \)(2S) \(\rightarrow \pi ^+ \pi ^- \)J/\(\psi \) decay distributions. Phys. Rev. D 62, 032002 (2000). https://doi.org/10.1103/PhysRevD.62.032002, arXiv:hep-ex/9909038
Beneke, M., Maltoni, F., Rothstein, I.Z.: QCD analysis of inclusive B decay into charmonium. Phys. Rev. D 59, 054003 (1999). https://doi.org/10.1103/PhysRevD.59.054003, arXiv:hep-ph/9808360
Beneke, M., Schuler, G.A., Wolf, S.: Quarkonium momentum distributions in photoproduction and B decay. Phys. Rev. D 62, 034004 (2000). https://doi.org/10.1103/PhysRevD.62.034004, arXiv:hep-ph/0001062
CLEO Collaboration: Inclusive decays of B mesons to charmonium. Phys. Rev. D 52, 2661 (1995). https://doi.org/10.1103/PhysRevD.52.2661
BaBar Collaboration: Study of inclusive production of charmonium mesons in B decay. Phys. Rev. D 67, 032002 (2003). https://doi.org/10.1103/PhysRevD.67.032002, arXiv:hep-ex/0207097
CDF Collaboration: Measurement of J/\(\psi \) and \(\psi \)(2S) polarization in p\(\bar{{\rm p}}\) collisions at \(\sqrt{s} = 1.8\) TeV. Phys. Rev. Lett. 85, 2886 (2000). https://doi.org/10.1103/PhysRevLett.85.2886, arXiv:hep-ex/0004027
Fleming, S., Hernandez, O.F., Maksymyk, I., Nadeau, H.: NRQCD matrix elements in polarization of J/\(\psi \) produced from \(b\) decay. Phys. Rev. D 55, 4098 (1997). https://doi.org/10.1103/PhysRevD.55.4098, arXiv:hep-ph/9608413
Krey, V., Balaji, K.R.S.: Polarized J/\(\psi \) production from B mesons at the Tevatron. Phys. Rev. D 67, 054011 (2003). https://doi.org/10.1103/PhysRevD.67.054011, arXiv:hep-ph/0209135
Faccioli, P., Lourenço, C., Seixas, J., Wöhri, H.K.: Determination of \(\chi _c\) and \(\chi _b\) polarizations from dilepton angular distributions in radiative decays. Phys. Rev. D 83, 096001 (2011). https://doi.org/10.1103/PhysRevD.83.096001, arXiv:1103.4882
ATLAS Collaboration: Measurement of the \(Z/\gamma ^*\) boson transverse momentum distribution in pp collisions at \(\sqrt{s} = 7\) TeV with the ATLAS detector. JHEP 09, 145 (2014). https://doi.org/10.1007/JHEP09(2014)145, arXiv:1406.3660
CMS Collaboration: Measurement of the transverse momentum spectra of weak vector bosons produced in proton-proton collisions at \(\sqrt{s}=8\) TeV. JHEP 02, 096 (2017). https://doi.org/10.1007/JHEP02(2017)096, arXiv:1606.05864
CMS Collaboration: Measurement of the Z boson differential cross section in transverse momentum and rapidity in proton-proton collisions at 8 TeV. Phys. Lett. B 749, 187 (2015). https://doi.org/10.1016/j.physletb.2015.07.065, arXiv:1504.03511
Guberina, B., Kühn, J.H., Peccei, R.D., Rückl, R.: Rare decays of the Z\(^0\). Nucl. Phys. B 174, 317 (1980). https://doi.org/10.1016/0550-3213(80)90287-4
Huang, T.-C., Petriello, F.: Rare exclusive decays of the Z-boson revisited. Phys. Rev. D 92, 014007 (2015). https://doi.org/10.1103/PhysRevD.92.014007, arXiv:1411.5924
Bodwin, G.T., Chung, H.S., Ee, J.-H., Lee, J.: Z-boson decays to a vector quarkonium plus a photon. Phys. Rev. D 97, 016009 (2018). https://doi.org/10.1103/PhysRevD.97.016009, arXiv:1709.09320
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Faccioli, P., Lourenço, C. (2023). Polarization in Cascade Decays. In: Particle Polarization in High Energy Physics. Lecture Notes in Physics, vol 1002. Springer, Cham. https://doi.org/10.1007/978-3-031-08876-6_6
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